Polynomial Fuzzy-model-based Control System Research Articles

Event-triggered control design with varying gains for polynomial fuzzy systems against DoS attacks

This paper presents an innovative event-triggered control scheme for addressing the stabilization problem of polynomial fuzzy systems under the influence of Denial-of-Service (DoS) attacks. The proposed controller utilizes a sampling-based event-triggered mechanism to reduce communication resources and avoid Zeno behavior. Furthermore, a novel polynomial fuzzy model-based control system is developed to investigate the impact of periodic DoS attacks and the addressed event-triggered mechanism on system stability. To improve system performance, control gains are updated at each triggering instant. The exponential stability criteria are formulated in the form of sum-of-square constraints, supported by a triggering instant dependent piecewise Lyapunov-Krasovskii functional and an online asynchronous premise reconstruction approach. Finally, the efficiency and usefulness of the theoretical findings are validated through simulation examples.

Stability Analysis of Positive Polynomial Fuzzy-Model-Based Control Systems With Time Delay Under Imperfect Premise Matching

This paper deals with the stability and positivity analysis of polynomial-fuzzy-model-based (PFMB) control systems with time delay, which is formed by a polynomial fuzzy model and a polynomial fuzzy controller connected in a closed loop, under imperfect premise matching. To improve the design and realization flexibility, the polynomial fuzzy model and the polynomial fuzzy controller are allowed to have their own set of premise membership functions. A sum-of-squares-based stability analysis approach using the Lyapunov stability theory is employed to investigate the positivity and stability of the PFMB control systems and synthesize the polynomial fuzzy controller. In order to relax the stability results, we propose two methods: first, membership functions are considered as symbolic variables in the stability analysis; and second, the property of the membership functions and the boundary information of the membership functions are considered in the stability analysis. A simulation example is given to illustrate the effectiveness of the proposed approach.

Membership‐Function‐Dependent Stability Analysis of Interval Type‐2 Polynomial Fuzzy‐Model‐Base Control Systems

In this paper, the stability analysis for interval type-2 (IT2) polynomial fuzzy-model-based (PFMB) control system using the information of membership functions is investigated. To tackle uncertainties, IT2 membership functions are used in the IT2 polynomial fuzzy model and IT2 polynomial fuzzy controller. The stability of IT2 PFMB control system is investigated based on the Lyapunov stability theory and both sets of membership function independent (MFI) and membership function dependent (MFD) stability conditions are derived on the basis of the sum-of-squares (SOS) approach. To make the stability conditions MFD, the boundary information of IT2 membership functions is used in the stability analysis. To extract richer information of IT2 membership functions, the operating domain is partitioned into sub-domains. In each sub-domain, the boundary information of IT2 membership functions and those of the upper and lower membership function are obtained. Furthermore, to further relax the conservativeness, a switching polynomial fuzzy controller, together with the informations obtained in each sub-domain, is employed in investigating the stability analysis. Numerical examples and simulation results are given to demonstrate the validity of MFD and MFD switching methods.

Relaxed stability conditions for polynomial‐fuzzy‐model‐based control system with membership function information

This study presents a new approach for stability analysis of polynomial-fuzzy-model-based (PFMB) control system using membership function information. For the purpose of extracting the regional information of the membership functions, the operating domain is partitioned into several sub-domains. In each sub-domain, the boundaries of every single membership function overlap term and the numerical relation among all the membership function overlap terms are represented as a group of inequalities. Through the S-procedure, the regional membership function information is taken into account in the stability analysis to relax the stability conditions. The operating domain partition scheme naturally arises the motivation of constructing the PFMB control system with the sub-domain fuzzy controllers. Each polynomial fuzzy controller works in its corresponding sub-domain such that the compensation capability of controller is enhanced. The sum-of-squares (SOS) approach is proposed to obtain the stability conditions of the PFMB control system using the Lyapunov stability theory. The PFMB control system studied in this study has the feature that the number of fuzzy rules and the membership function shapes of the polynomial fuzzy controller can be designed independently from the polynomial fuzzy model. To verify the stability analysis result, a numerical example is given to demonstrate the validity of the proposed method.

Stabilization of Interval Type-2 Polynomial-Fuzzy-Model-Based Control Systems

In this paper, the stability of polynomial-fuzzy-model-based (PFMB) systems equipped with mismatched interval type-2 (IT2) membership functions is investigated. Unlike the membership-function-independent methods, the information and properties of IT2 membership functions are considered in the stability analysis and contained in the stability conditions in terms of sum-of-squares (SOS) based on the Lyapunov stability theory. Three methods, demonstrating their own advantages, are proposed to conduct the stability analysis for the IT2 PFMB control systems. In the first one, we divide the operating domain into subdomains and then conduct the stability analysis incorporating the information and properties of the IT2 membership functions in subdomains. Through this approach, the stability conditions can be further relaxed compared with the membership-function-independent analysis. Polynomial functions are adopted in the second method to approximate the IT2 membership functions. The advantage of this method compared with the first one is that richer information of IT2 membership functions is considered without increasing the number of SOS conditions. In the third one, we combine the advantages of both the first and the second method offering a new approach which utilizes the information and properties of the lower and upper IT2 membership functions in subdomains through simpler polynomial approximation functions. It can be shown that more relaxed stability conditions can be obtained compared with the first two methods. Numerical examples and simulations are presented to verify the effectiveness of the proposed methods.

Two-Step Stability Analysis for General Polynomial-Fuzzy-Model-Based Control Systems

This paper investigates the stability of a polynomial-fuzzy-model-based (PFMB) control system formed by a nonlinear plant represented by a polynomial fuzzy model and a polynomial fuzzy controller connected in a closed loop. Three cases of polynomial fuzzy controllers are proposed for the control process with the consideration of a matched/mismatched number of rules and/or premise membership functions, which demonstrate different levels of controller complexity, design flexibility, and stability analysis results. A general polynomial Lyapunov function candidate is proposed to investigate the system stability. Unlike the published work, there is no constraint on the polynomial Lyapunov function candidate, which is independent of the form of the polynomial fuzzy model. Thus, it can be applied to a wider class of PFMB control systems and potentially produces more relaxed stability analysis results. Two-step stability conditions in terms of sum-of-squares (SOS) are obtained to numerically find a feasible solution. To facilitate the stability analysis and relax the stability analysis result, the boundary information of membership functions is taken into account in the stability analysis and incorporated into the SOS-based stability conditions. Simulation examples are given to illustrate the effectiveness of the proposed approach.

Stability Analysis of Polynomial-Fuzzy-Model-Based Control Systems With Mismatched Premise Membership Functions

This paper investigates the stability of polynomial-fuzzy-model-based (PFMB) control system, which is formed by a polynomial fuzzy model and a polynomial fuzzy controller connected in a closed loop. To enhance the design flexibility, the number of rules and the shape of premise membership functions of the polynomial fuzzy controller are considered to be chosen freely and are different from those of the polynomial fuzzy model, however, which make the stability analysis more difficult and potentially lead to conservative stability analysis result. A sum-of-squares (SOS)-based stability analysis approach using the Lyapunov stability theory is proposed to investigate the stability of the PFMB control systems and synthesize the polynomial fuzzy controller. To facilitate the stability analysis and relax the stability analysis result, the property of the membership functions and the boundary information of the membership grades and premise variables are taken into account in the stability analysis and incorporated into the SOS-based stability conditions. A simulation example is given to illustrate the effectiveness of the proposed approach.

Analysis of stability and robust stability of polynomial fuzzy model-based control systems using a sum-of-squares approach

The well known Takagi---Sugeno (T---S) fuzzy model can be extended in different ways including the polynomial fuzzy model, whose consequent parts are polynomial sub-systems. Compared with the traditional T---S fuzzy model, the polynomial fuzzy model can represent a nonlinear system more accurately with a smaller number of fuzzy logic rules. It is worth emphasizing that the stability analysis and controller design of polynomial fuzzy model-based (PFMB) control systems are not based on the linear matrix inequalities but the recently developed sum-of-squares decompositions. In this paper, based on an existing result for traditional fuzzy control systems, we propose a new stability condition for the stability analysis of PFMB control systems. Furthermore, the stability of PFMB control systems with parameter uncertainties is investigated. The popular inverted pendulum and an unstable nonlinear system are employed to demonstrate the quality of the proposed stability conditions.

Output Regulation of Polynomial-Fuzzy-Model-Based Control Systems

This paper investigates the output regulation problem of polynomial-fuzzy-model-based (PFMB) control systems with the consideration of system stability and control synthesis using the sum-of-squares (SOS)-based approach. A PFMB control system is formed by a polynomial fuzzy model and a polynomial fuzzy controller with integral action connected in a closed loop. Three cases of PFMB control systems regarding the premise membership functions and number of fuzzy rules, which have their own properties and advantages, are considered. The stability of the PFMB control systems is investigated based on the Lyapunov stability theory. With consideration of the information of membership functions, SOS-based stability conditions corresponding to each case are obtained to guarantee system stability and to realize output regulation. Simulation examples are given to compare the three cases and illustrate the effectiveness of the proposed approach.

Polynomial Fuzzy-Model-Based Control Systems: Stability Analysis Via Piecewise-Linear Membership Functions

This paper presents the stability analysis of polynomial fuzzy-model-based (PFMB) control systems using the sum-of-squares (SOS) approach. The PFMB control system under consideration requires that the polynomial fuzzy model and polynomial fuzzy controller share neither the same premise membership functions nor the same number of fuzzy rules. This class of PFMB control systems offers a greater design flexibility to the polynomial fuzzy controller. However, due to the imperfectly matched membership functions, it usually produces more conservative stability conditions by following the traditional stability-analysis approach for the FMB control systems. To facilitate the stability analysis, piecewise-linear membership functions (PLMFs) are proposed, which offer a nice property that the grades of membership are governed by a finite number of sample points. Thus, it allows the PLMFs to be brought to the SOS-based stability conditions, which are applied to the PFMB control systems with the specified PLMFs rather than any shapes. The system stability can be examined by checking only the PFMB control system at the sample points. It is worth mentioning that the PLMFs, which are not necessarily implemented physically, are a mathematical tool to carry out the stability analysis. To verify the stability-analysis result, a simulation example is given to demonstrate the effectiveness of the proposed approach.